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Published January 10, 2025 | Version Published
Journal Article Open

The NANOGrav 15 yr Data Set: Running of the Spectral Index

Creators

  • 1. ROR icon University of Wisconsin–Milwaukee
  • 2. ROR icon Newcastle University
  • 3. ROR icon Goddard Space Flight Center
  • 4. ROR icon Oregon State University
  • 5. ROR icon Widener University
  • 6. ROR icon University of Florida
  • 7. ROR icon Cornell University
  • 8. ROR icon University of Birmingham
  • 9. ROR icon West Virginia University
  • 10. ROR icon University of Connecticut
  • 11. ROR icon Vanderbilt University
  • 12. ROR icon New Mexico Institute of Mining and Technology
  • 13. ROR icon Montana State University
  • 14. ROR icon Franklin & Marshall College
  • 15. ROR icon National Academy of Sciences
  • 16. ROR icon United States Naval Research Laboratory
  • 17. ROR icon University of British Columbia
  • 18. ROR icon George Mason University
  • 19. ROR icon National Radio Astronomy Observatory
  • 20. ROR icon Hillsdale College
  • 21. ROR icon Eureka Scientific
  • 22. ROR icon University of Münster
  • 23. ROR icon University of Maryland, College Park
  • 24. ROR icon University of California, Berkeley
  • 25. ROR icon University of Montana
  • 26. ROR icon University of Michigan–Ann Arbor
  • 27. ROR icon California Institute of Technology
  • 28. ROR icon University of Washington
  • 29. ROR icon Search for Extraterrestrial Intelligence
  • 30. ROR icon Rochester Institute of Technology
  • 31. ROR icon Yale University
  • 32. ROR icon Jet Propulsion Lab
  • 33. ROR icon State University of New York at Oswego
  • 34. ROR icon National Centre for Nuclear Research
  • 35. ROR icon University of Toronto
  • 36. Deceased.
  • 37. Green Bank Observatory, P.O. Box 2, Green Bank, WV 24944, USA
  • 38. ROR icon University of the Pacific
  • 39. ROR icon University of Hull
  • 40. ROR icon International Centre for Radio Astronomy Research
  • 41. ROR icon Deutsches Elektronen-Synchrotron DESY
  • 42. ROR icon Lafayette College
  • 43. ROR icon Carnegie Observatories
  • 44. ROR icon Tufts University
  • 45. ROR icon Eötvös Loránd University
  • 46. ROR icon Arecibo Observatory
  • 47. ROR icon Texas Tech University
  • 48. ROR icon University of Puerto Rico System
  • 49. ROR icon Pennsylvania State University
  • 50. Giant Army, 915A 17th Avenue, Seattle, WA 98122, USA
  • 51. ROR icon University of Colorado Boulder
  • 52. ROR icon Harvard-Smithsonian Center for Astrophysics
  • 53. ROR icon Oberlin College
  • 54. ROR icon Max Planck Institute for Gravitational Physics
  • 55. ROR icon Middle East Technical University
  • 56. ROR icon Ben-Gurion University of the Negev
  • 57. ROR icon Feza Gürsey Institute
  • 58. ROR icon Northwestern University
  • 59. ROR icon Adler Planetarium

Abstract

The NANOGrav 15 yr data provide compelling evidence for a stochastic gravitational-wave (GW) background at nanohertz frequencies. The simplest model-independent approach to characterizing the frequency spectrum of this signal consists of a simple power-law fit involving two parameters: an amplitude A and a spectral index γ. In this Letter, we consider the next logical step beyond this minimal spectral model, allowing for a running (i.e., logarithmic frequency dependence) of the spectral index, γ_(run)(f) = γ+βln(f/f_(ref)). We fit this running-power-law (RPL) model to the NANOGrav 15 yr data and perform a Bayesian model comparison with the minimal constant-power-law (CPL) model, which results in a 95% credible interval for the parameter β consistent with no running, β ∈ [−0.80, 2.96], and an inconclusive Bayes factor, B(RPL versus CPL) = 0.69±0.01. We thus conclude that, at present, the minimal CPL model still suffices to adequately describe the NANOGrav signal; however, future data sets may well lead to a measurement of nonzero β. Finally, we interpret the RPL model as a description of primordial GWs generated during cosmic inflation, which allows us to combine our results with upper limits from Big Bang nucleosynthesis, the cosmic microwave background, and LIGO–Virgo–KAGRA.

Copyright and License

© 2025. The Author(s). Published by the American Astronomical Society.

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Acknowledgement

L.B. acknowledges support from the National Science Foundation under award AST-1909933 and from the Research Corporation for Science Advancement under Cottrell Scholar Award No. 27553. P.R.B. is supported by the Science and Technology Facilities Council, grant No. ST/W000946/1. S.B. gratefully acknowledges the support of a Sloan Fellowship and the support of NSF under award #1815664. M.C. and S.R.T. acknowledge support from NSF AST-2007993. M.C. was supported by the Vanderbilt Initiative in Data Intensive Astrophysics (VIDA) Fellowship. Support for this work was provided by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc./National Radio Astronomy Observatory. Pulsar research at UBC is supported by an NSERC Discovery Grant and by CIFAR. K.C. is supported by a UBC Four Year Fellowship (6456). M.E.D. acknowledges support from the Naval Research Laboratory by NASA under contract S-15633Y. T.D. and M.T.L. are supported by an NSF Astronomy and Astrophysics Grant (AAG) award number 2009468. E.C.F. is supported by NASA under award number 80GSFC21M0002. G.E.F., S.C.S., and S.J.V. are supported by NSF award PHY-2011772. K.A.G. and S.R.T. acknowledge support from an NSF CAREER award #2146016. A.D.J. and M.V. acknowledge support from the Caltech and Jet Propulsion Laboratory President's and Director's Research and Development Fund. A.D.J. acknowledges support from the Sloan Foundation. The work of N.La., X.S., and D.W. is partly supported by the George and Hannah Bolinger Memorial Fund in the College of Science at Oregon State University. N.La. acknowledges the support from the Larry W. Martin and Joyce B. O'Neill Endowed Fellowship in the College of Science at Oregon State University. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). R.R.L.d.S. is supported in part by the National Science Centre (Poland) under the research grant No. 2020/38/E/ST2/00126. D.R.L. and M.A.M. are supported by NSF #1458952. M.A.M. is supported by NSF #2009425. C.M.F.M. was supported in part by the National Science Foundation under grant Nos. NSF PHY-1748958 and AST-2106552. A.Mi. is supported by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy–EXC 2121 Quantum Universe–390833306. The Dunlap Institute is funded by an endowment established by the David Dunlap family and the University of Toronto. K.D.O. was supported in part by NSF grant No. 2207267. T.T.P. acknowledges support from the Extragalactic Astrophysics Research Group at Eötvös Loránd University, funded by the Eötvös Loránd Research Network (ELKH), which was used during the development of this research. H.A.R. is supported by NSF Partnerships for Research and Education in Physics (PREP) award No. 2216793. S.M.R. and I.H.S. are CIFAR Fellows. Portions of this work performed at NRL were supported by ONR 6.1 basic research funding. J.D.R. also acknowledges support from start-up funds from Texas Tech University. The work of K.Sc., T.S., and R.v.E. is supported by Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group (Graduiertenkolleg) 2149: Strong and Weak Interactions–from Hadrons to Dark Matter. J.S. is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2202388 and acknowledges previous support by the NSF under award 1847938. C.U. acknowledges support from BGU (Kreitman fellowship) and the Council for Higher Education and Israel Academy of Sciences and Humanities (Excellence fellowship). C.A.W. acknowledges support from CIERA, the Adler Planetarium, and the Brinson Foundation through a CIERA-Adler postdoctoral fellowship. O.Y. is supported by the National Science Foundation Graduate Research Fellowship under grant No. DGE-2139292.

Contributions

This Letter uses over a decade's worth of pulsar timing observations and is the product of the work of many people. K.Sc. initiated and led the project, designed the analysis, carried out the analytical calculations, and wrote the manuscript. R.R.L.d.S. carried out the MCMC analysis and prepared all figures. D.E. cross-checked parts of the MCMC analysis. R.v.E. and T.Sc. cross-checked and evaluated the parameter bounds in Section 4. The authors thank Andrew Casey-Clyde for suggesting additional references, Timothy Dolch and Ken Olum for helpful feedback and comments on the manuscript, and all members of the NANOGrav New Physics Working Group who provided feedback and comments in working group meetings.

Data Availability

G.A., A.A., A.M.A., Z.A., P.T.B., P.R.B., H.T.C., K.C., M.E.D., P.B.D., T.D., E.C.F., W.F., E.F., G.E.F., N.G.D., D.C.G., P.A.G., J.G., R.J.J., M.L.J., D.L.K., M.K., M.T.L., D.R.L., J.L., R.S.L., A.M., M.A.M., N.M., B.W.M., C.N., D.J.N., T.T.N., B.B.P.P., N.S.P., H.A.R., S.M.R., P.S.R., A.S., C.S., B.J.S.A., I.H.S., K.St., A.S., J.K.S., and H.M.W. developed timing models and ran observations for the NG15 data set.

Facilities

Part of this work was conducted using the High Performance Computing Cluster PALMA II at the University of Münster. The use of the CIS computer cluster at the National Centre for Nuclear Research in Warsaw is gratefully acknowledged.

Funding

The NANOGrav Collaboration receives support from National Science Foundation (NSF) Physics Frontiers Center award Nos. 1430284 and 2020265, the Gordon and Betty Moore Foundation, NSF AccelNet award No. 2114721, an NSERC Discovery Grant, and CIFAR. The Arecibo Observatory is a facility of the NSF operated under cooperative agreement (AST-1744119) by the University of Central Florida (UCF) in alliance with Universidad Ana G. Méndez (UAGM) and Yang Enterprises (YEI), Inc. The Green Bank Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc. The National Radio Astronomy Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc.

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Additional details

Related works

Is new version of
Discussion Paper: arXiv:2408.10166 (arXiv)
Is supplemented by
Other: https://www.uni-muenster.de/IT/HPC (URL)

Funding

National Science Foundation
AST-1909933
Research Corporation for Science Advancement
27553
Science and Technology Facilities Council
ST/W000946/1
Alfred P. Sloan Foundation
National Science Foundation
AST-1815664
National Science Foundation
AST-2007993
Vanderbilt University
National Science Foundation
Grote Reber Fellowship Program
Natural Sciences and Engineering Research Council
Canadian Institute for Advanced Research
University of British Columbia
United States Naval Research Laboratory
S-15633Y
National Science Foundation
AST-2009468
National Aeronautics and Space Administration
80GSFC21M0002
National Science Foundation
PHY-2011772
National Science Foundation
PHY-2146016
California Institute of Technology
President's Research and Development Fund
Jet Propulsion Laboratory
Director's Research and Development Fund
Oregon State University
National Aeronautics and Space Administration
80NM0018D0004
National Science Center
2020/38/E/ST2/00126
National Science Foundation
OIA-1458952
National Science Foundation
AST-2009425
National Science Foundation
PHY-1748958
National Science Foundation
AST-2106552
Deutsche Forschungsgemeinschaft
EXC 2121 — 390833306
University of Toronto
Dunlap Institute for Astronomy and Astrophysics
National Science Foundation
PHY-2207267
Eötvös Loránd University
National Science Foundation
PHY-2216793
Office of Naval Research
6.1 Basic Research Funding
Texas Tech University
Deutsche Forschungsgemeinschaft
2149
National Science Foundation
AST-2202388
National Science Foundation
AST-1847938
Ben-Gurion University of the Negev
Kreitman Fellowship
Council for Higher Education
Israel Academy of Sciences and Humanities
Northwestern University
Center for Interdisciplinary Exploration and Research in Astrophysics
Adler Planetarium
Brinson Foundation
National Science Foundation
DGE-2139292
National Science Foundation
PHY-1430284
National Science Foundation
PHY-2020265
Gordon and Betty Moore Foundation
National Science Foundation
OISE-2114721
National Science Foundation
AST-1744119
University of Central Florida
Ana G. Mendez University System

Dates

Accepted
2024-11-28
Accepted
Available
2025-01-06
Published

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Published